In this paper, we propose a modified version of Picard type iterative algorithm for finding a fixed point of a nonexpansive mapping defined on a closed convex subset of a Hilbert space. We prove the strong convergence of the sequence generated by the proposed algorithm to a fixed point of a nonexpensive map, such fixed point is also a solution of a variational inequality. As a particular case, we derive an algorithm for computing the approximate solutions of the constrained convex minimization problem. We illustrate our results by some examples. The results of this paper extend and improve several known results in the literature.